Question: Let $S$ be a surface in 3D described by the equation $4x^2 + 4y^2 + z^2 = 9$. Fill in the rest of the equation of the plane tangent to $S$ at $(1, 1, 1)$.
The equation for a tangent plane of an implicitly defined surface $F(x, y, z) = 0$ at the point $(a, b, c)$ is: $F_x(x - a) + F_y(y - b) + F_z(z - c) = 0$ [What's the intuition behind the formula?] We can see from the formula that the two values we're missing are $F_x$ and $F_y$. $\begin{aligned} F_x &= 8x = 8 \\ \\ F_y &= 8y = 8 \end{aligned}$ Here's the completed equation for the tangent plane of $S$ at $(1, 1, \sqrt{2})$ : $8(x - 1) + 8(y - 1) + 2(z - 1) = 0$